Friðrik was born in Reykjavík, Iceland. A first-time winner of the Icelandic Championship in 1952 and of the Scandinavian Championship a year later, he rapidly became recognised as the strongest Icelandic player of his generation. Friðrik's first result of international note was his shared first with Viktor Korchnoi at Hastings 1955-56.

Among his best tournament results were joint third in the first Piatigorsky Cup, Los Angeles 1963, with 7.5/14 and shared first with Ljubomir Ljubojević at Wijk aan Zee 1976, ahead of Mikhail Tal. In 1978, he succeeded Max Euwe as President of the international chess governing body FIDE. During he tenure he presided over the 1981 Korchnoi-Karpov World Championship match. Since Korchnoi defected from the Soviet Union in 1976, the Soviets were holding Korchnoi's son, Igor. Olafsson delayed the planned September 19 start date of the match in a bid to get the Soviets to release Victor's son. For this attempt, Olafsson drew the wrath of the Soviets, who then backed the FIDE Vice-President, Florencio Campomanes for Presidency of FIDE. After losing the election in 1982, Olafsson was appointed Secretary to the Icelandic Parliament. Friðrik continued to play occasionally into the 21st century, winning a rapid match against fellow veteran Bent Larsen in 2003 by a score of 5-3.

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